Step 1 :

After factoring out $ -1 $ we have:

$$ -4x^{2}+8x-3 = - ~ ( 4x^{2}-8x+3 ) $$

Step 2 :

Step 2: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:

$$ a = 4 , b = -8 ~ \text{ and } ~ c = 3 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 4 }$ by the constant term $\color{blue}{c = 3} $.

$$ a \cdot c = 12 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 12 $ and add to $ b = -8 $.

Step 5: All pairs of numbers with a product of $ 12 $ are:

PRODUCT = 12
1     12	-1     -12
2     6	-2     -6
3     4	-3     -4

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -8 }$

PRODUCT = 12 and SUM = -8
1     12	-1     -12
2     6	-2     -6
3     4	-3     -4

Step 7: Replace middle term $ -8 x $ with $ -2x-6x $:

$$ 4x^{2}-8x+3 = 4x^{2}-2x-6x+3 $$

Step 8: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -3 $ out of the last two terms.

$$ 4x^{2}-2x-6x+3 = 2x\left(2x-1\right) -3\left(2x-1\right) = \left(2x-3\right) \left(2x-1\right) $$

Polynomial Factoring Calculator